First off, let’s get the whole jelly-doughnut fiction out of the way. For decades, people have been chuckling over the oft-repeated “fact” that when he delivered his now-famous speech at Berlin’s Schöneberg city hall in June 1963, John F. Kennedy flubbed the oration’s critical line. Instead of declaring his solidarity with the German people with a rousing, “I am a Berliner!” (Ich bin Berliner) — so the story goes — Kennedy instead proclaimed, “I am a jelly doughnut!” (Ich bin ein Berliner).

It’s a pretty good story, and it’s even more comical when it’s repeated, as it has been countless times in the subsequent decades, in JFK’s distinctive Boston accent and with his unique cadence. Alas, for comedians and for cocktail-party trivia experts everywhere, Kennedy’s assertion was not only perfectly comprehensible, but positively stirring, to the thousands of Germans who saw his speech live and to the millions of others who heard it on the radio or saw it on TV.

“I am a Berliner!” It might not be as hilarious as the apocryphal jelly-doughnut line — but five decades on, JFK’s simple declaration still feels inspiring.

Here, more than 50 years after Kennedy’s June 1963 speech in Berlin, LIFE.com recalls not only that one historic moment, but the look and the feel — the unprecedented energy— of his trip to Germany. Kennedy drew boisterous and, for the most part, adoring crowds wherever he traveled, and less than two decades after the end of World War II, in a West Germany that was now an American ally, was received as something of a rock star by young and old alike.

Staged less than five months before his assassination would stun the world, in a nation ripped apart by competing ideologies and by the brute, concrete symbol of the Cold War —the Berlin Wall— JFK’s triumphant German tour was one of the earliest and most poignant watersheds of the 1960s. As America moved deeper into the decade, and as violence seemed to erupt from every seam in the culture — the terrorist church bombing in Birmingham mere months later; the assassinations of Malcolm X, Dr. King and Robert Kennedy; the war in Vietnam; the Mansons; Altamont and on and on — the promise of a new, re-imagined and perhaps even morally ascendant United States flickered, and faded.